Filtrations of covers, Sheaves, and Integration Michael Robinson

Acknowledgements ● Collaborators: – Brett Jefgerson, Clifg Joslyn, Brenda Praggastis, Emilie Purvine (PNNL) – Chris Capraro, Grant Clarke, Griffjn Kearney, Janelle Henrich, Kevin Palmowski (SRC) – J Smart, Dave Bridgeland (Georgetown) ● Students: – Michael Rawson – Robby Green – Philip Coyle – Metin Toksoz-Exley – Samara Fantie – Fangei Lan – Jackson Williams ● Recent funding: DARPA, ONR, AFRL Michael Robinson

Key ideas ● Motivate fjltrations of partial covers as generalizing consistency fjltrations of sheaf assignments ● Explore fjltrations of partial covers as interesting mathematical objects in their own right ● Instill hope that fjltrations of partial covers may be (incompletely) characterized Big caveat: Only fjnite spaces are under consideration! Michael Robinson

Consistency fjltrations Michael Robinson

Context ● Assemble stochastic models of data locally into a global topological picture – Persistent homology is sensitive to outliers – Statistical tools are less sensitive to outliers, but cannot handle (much) global topological structure – Sheaves can be built to mediate between these two extremes... this is what I have tried to do for the past decade or so ● The output is the consistency fjltration of a sheaf assignment Michael Robinson

Topologizing a partial order Open sets are unions of up-sets Michael Robinson

Topologizing a partial order Intersections of up-sets are also up-sets Michael Robinson

A sheaf on a poset is... ℝ A set assigned to (0 1) ℝ each element, called (1 0) ℝ 2 a stalk , and … ℝ 2 ( ) ( ) -3 3 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 1 0 1 ℝ ℝ 2 ℝ 3 2 3 2 -2 1 3 -3 ( ) 0 1 1 1 -1 1 0 1 ( ) (1 -1) 0 1 1 0 ℝ 3 Stalks can be measure spaces! ℝ 2 We can handle stochastic data This is a sheaf of vector spaces on a partial order Michael Robinson

A sheaf on a poset is... ℝ … restriction functions (0 1) between stalks, ℝ (1 0) following the ℝ 2 ℝ 2 order relation… ( ) ( ) -3 3 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 1 0 1 ℝ ℝ 2 ℝ 3 2 3 2 -2 1 3 -3 ( ) 0 1 1 1 -1 1 0 1 ( ) (1 -1) 0 1 (“Restriction” 1 0 because it goes from ℝ 3 bigger up-sets to smaller ones) ℝ 2 This is a sheaf of vector spaces on a partial order Michael Robinson

A sheaf on a poset is... … so that the diagram ℝ commutes! (0 1) ℝ (1 0) ( ) = (0 1) ( ) 0 1 1 1 0 1 (1 0) ℝ 2 1 0 1 0 1 1 ℝ 2 ( ) ( ) -3 3 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 ( ) 2 -2 ( ) ( ) 1 0 1 -3 3 0 1 1 0 1 1 -1 = 3 -3 0 1 1 -4 4 1 0 ℝ ℝ 2 ℝ 3 2 3 2 -2 2 2 -2 1 (1 -1) = 3 -3 3 3 -3 ( ) 0 1 1 1 -1 1 1 -1 1 0 1 ( ) (1 -1) 0 1 1 0 ℝ 3 ℝ 2 This is a sheaf of vector spaces on a partial order Michael Robinson

An assignment is... (-4) … the selection of a (0 1) (-4) value on some open (1 0) ( ) -3 sets ( ) -4 -4 ( ) ( ) -3 -3 3 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 1 0 1 ( ) -2 3 -3 (-1) 2 2 -1 3 2 -2 1 3 -3 ( ) 0 1 1 1 -1 1 0 1 ( ) (1 -1) 0 1 1 0 2 ( ) 3 2 0 3 The term serration is more common, but perhaps more opaque. Michael Robinson

A global section is... (-4) … an assignment (0 1) (-4) that is consistent (1 0) ( ) -3 with the restrictions ( ) -4 -4 ( ) ( ) -3 -3 3 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 1 0 1 ( ) -2 3 -3 (-1) 2 2 -1 3 2 -2 1 3 -3 ( ) 0 1 1 1 -1 1 0 1 ( ) (1 -1) 0 1 1 0 2 ( ) 3 2 0 3 Michael Robinson

Some assignments aren’t consistent (-4) … but they might (0 1) (-4) be partially (1 0) ( ) -3 consistent ( ) -4 -4 ( ) ( ) -3 -3 3 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 1 0 1 ( ) -2 3 -3 (+1) 2 2 -1 3 2 -2 1 3 -3 ( ) 0 1 1 1 -1 1 0 1 ( ) (1 -1) 0 1 1 0 2 ( ) 3 2 1 3 Michael Robinson

Consistency radius is... (-4) … the maximum (or some other norm) (0 1) (-4) (1 0) distance between the ( ) -3 ( ) value in a stalk and -4 -4 ( ) ( ) -3 -3 3 the values 1 0 1 (-2 1) ( ) -4 4 0 1 1 0 1 1 propagated 1 0 1 ( ) ( ) 2 ( ) along the -2 3 3 0 1 1 - = 2 3 -3 (+1) 2 2 restrictions 1 0 1 2 1 -1 3 2 -2 1 3 -3 ( ) 0 1 1 1 -1 2 -2 1 0 1 ( ) (1 -1) (+1) - 3 -3 = 2 14 0 1 MAX ≥ 2 14 1 -1 1 0 2 ( ) 3 2 1 ( ) 3 2 (1 -1) - 1 = 2 3 Note: lots more restrictions to check! Michael Robinson

Amateur radio foxhunting Typical sensors: ● Bearing to Fox ● Fox signal strength ● GPS location Michael Robinson

Bearing sensors 9 0 1 1 2 0 6 0 1 5 0 3 0 0 . 5 1 8 0 0 2 1 0 3 3 0 2 4 0 3 0 0 2 7 0 Antenna pattern Michael Robinson

Bearing sensors… reality… 9 0 1 1 2 0 6 0 1 5 0 3 0 0 . 5 1 8 0 0 2 1 0 3 3 0 2 4 0 3 0 0 2 7 0 Antenna pattern Michael Robinson

Bearing observations Bearing as a function of sensor position 9 0 1 1 2 0 6 0 Recenter 1 5 0 3 0 0 . 5 1 8 0 0 2 1 0 3 3 0 2 4 0 3 0 0 2 7 0 Antenna pattern Michael Robinson

Bearing sheaf Typical M bearing function: 9 0 1 1 2 0 6 0 1 5 0 3 0 0 . 5 1 8 0 0 2 1 0 3 3 0 2 4 0 3 0 0 2 7 0 Antenna pattern Michael Robinson

Bearing sheaf (two sensors) Sensor 1 position, Bearing Fox position Sensor 2 position, Bearing ℝ 2 × S 1 ℝ 2 ℝ 2 × S 1 pr 1 (pr 2 , M bearing ) pr 1 (pr 2 , M bearing ) ℝ 2 ×ℝ 2 ℝ 2 ×ℝ 2 Fox position, Sensor 1 position Fox position, Sensor 2 position Global sections of this sheaf correspond to two bearings whose sight lines intersect at the fox transmitter Michael Robinson

Consistency of proposed fox locations Consistency radius minimization … … does not converge! … converges to a likely fox location Michael Robinson

Local consistency radius Consistency radius of this open set = 0 ½ { C } (½) (1) 0 1 { A,C } { B,C } (1) (2) (1) ⅓ { A , B , C } Lemma: Consistency radius on an open set U is computed by only considering open sets V 1 ⊆ V 2 ⊆ U Michael Robinson

Local consistency radius Consistency radius of this open set = 0 ½ (½) (1) c( U ) = ½ 0 1 (1) (2) (1) ⅓ Lemma: Consistency radius does not decrease as its support grows: if U ⊆ V then c ( U ) ≤ c ( V ). Michael Robinson

Local consistency radius c( U ∩ V ) = 0 ½ (½) (1) c( U ) = ½ c( V ) = ½ 0 1 (1) (2) (1) ⅓ Lemma: Consistency radius does not decrease as its support grows: if U ⊆ V then c ( U ) ≤ c ( V ). Michael Robinson

Consistency radius is not a measure c( U ∩ V ) = 0 ½ (½) (1) c( U ) = ½ c( V ) = ½ 0 1 (1) (2) (1) ⅓ c( U ∪ V ) = ⅔ ≠ c( U ) + c( V ) – c( U ∩ V ) (Consistency radius yields an inner measure after some work) Michael Robinson

The consistency fjltration ● … assigns the set of open sets (open cover) with consistency less than a given threshold ● Lemma: consistency fjltration is itself a sheaf of collections of open sets on (ℝ,≤). Restrictions in this sheaf are cover coarsenings . ½ ½ (½) (1) 1 ½ refjne refjne (½) (1) (1) (½) (1) 0 1 (1) 0 1 (1) 0 1 (1) (1) ⅓ (2) (1) ⅓ (2) (1) ⅓ (2) (1) ⅓ Consistency radius = ⅔ Consistency threshold 0 ½ ⅔ Michael Robinson

Filtrations of partial covers Michael Robinson

Covers of topological spaces ● Classic tool: Čech cohomology – Coarse – Usually blind to the cover; only sees the underlying space ● Cover measures (Purvine, Pogel, Joslyn, 2017) – How fjne is a cover? – How overlappy is a cover? Michael Robinson

Cover measures ● Theorem: (Birkhofg) The set of covers ordered by refjnement has an explicit rank function – The rank of a given cover is the number of sets in its downset as an antichain of the Boolean lattice – This counts the number of sets of consistent faces there are ● Conclusion: An assignment whose maximal cover has a higher rank is more self-consistent Michael Robinson

Cover measures ● Consider the following two covers of {1,2,3,4} A B 1 2 3 4 1 2 3 4 Refjne Refjne 1 2 3 4 1 2 3 4 Since 6 < 11, Total = 6 sets cover B is coarser Refjne 1 2 3 4 Total = 11 sets Michael Robinson

The lattice of covers ● Theorem: The lattice of Coarser covers is graded using this rank function 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Finer Lattice graphic by E. Purvine Michael Robinson

Defjning CTop : partial covers ● Start with a fjxed topological space ● Objects: Collections of open sets ● No requirement of coverage 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Michael Robinson

Defjning CTop : partial covers ● Morphisms are r efjnements of covers: If 𝒱 and 𝒲 are partial 1 2 3 4 5 6 covers, 𝒲 refjnes 𝒱 if for all V in 𝒲 there is a U in 𝒱, with Refjne V ⊆ U . 1 2 3 4 5 6 ● Convention: 𝒱 → 𝒲 Refjne 1 2 3 4 5 6 Michael Robinson

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